VOLUMES OF RESTRICTED MINKOWSKI SUMS AND THE FREE ANALOG OF THE ENTROPY POWER INEQUALITY

In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutat ive theory: free probability theory (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannon's entropy power inequal ity ([6, 1]) has an analogue for the free entropy chi(X) (Theorem 2.1) . The free entropy, consistent with Boltzmann's formula S = k log W, w as defined via volumes of matricial microstates. Proving the free entr opy power inequality naturally becomes a geometric question. Restricti ng the Minkowski sum of two sets means to specify the set of pairs of points which will be added. The relevant inequality, which holds when the set of addable points is sufficiently large, differs from the Brun n-Minkowski inequality by having the exponent 1/n replaced by 2/n. Its proof uses the rearrangement inequality of Brascamp-Lieb-Luttinger ([ 2]). Besides the free entropy power inequality, note that the inequali ty for restricted Minkowski sums may also underlie the classical Shann on entropy power inequality (see 3.2 below).